This book focuses on elementary and middle school children’s understanding of mathematics as well as the cognitive aspects involved in the development of mathematical knowledge, skills, and understanding. Children’s success in and understanding of mathematics stem from factors beyond the mathematics curriculum. Researchers are increasingly becoming aware of the necessity to consider a complex set of variables when accounting for large individual differences in mathematics achievement. These chapters contribute to how both researchers and educators can consider the multidimensionality of skills involved in developing mathematical knowledge in the middle school years as well as to how this knowledge can be used to enhance practices in the mathematics classroom. Topics include the cognitive and spatial skills involved in mathematics knowledge, the role of motivation in mathematics learning, the neurological processes and development of children’s mathematics skills, the development of understanding of arithmetic and fraction concepts, the factors relating to children’s word problem success, and techniques to promote mathematics understanding.
This book and its companion, Mathematical Teaching and Learning, take an interdisciplinary perspective to mathematical learning and development in the elementary and middle school years. The authors and perspectives in this book draw from education, neuroscience, developmental psychology, and cognitive psychology. The book will be relevant to scholars/educators in the field of mathematics education and also those in childhood development and cognition. Each chapter also includes practical tips and implications for parents as well as for educators and researchers.
Author(s): Katherine M. Robinson, Adam K. Dubé, Donna Kotsopoulos
Publisher: Springer
Year: 2023
Language: English
Pages: 284
City: Cham
Contents
Contributors
Abbreviations
Chapter 1: An Introduction to Mathematical Cognition and Understanding in the Elementary and Middle School Years
1.1 Introduction
References
Part I: Cognitive Factors
Chapter 2: Infusing Spatial Thinking Into Elementary and Middle School Mathematics: What, Why, and How?
2.1 Introduction
2.2 What Is Spatial Thinking?
2.3 Why Are Spatial Skills and Mathematics Related?
2.4 Does Spatial Instruction Improve Mathematics Performance?
2.5 How to Best Leverage the Space-Mathematics Association?
2.5.1 Isolated Approaches to Spatial Training
2.5.2 Embedded Approaches to Spatial Training
2.5.3 Strengths, Limitations, and Theoretical Underpinnings
2.6 Translating Theory to Practice: Infusing Spatial Training Into Mathematics Teaching
2.7 Conclusion
References
Chapter 3: Understanding the Relationship Between Attention, Executive Functions, and Mathematics: Using a Function-Specific A...
3.1 Introduction
3.2 Attentional Abilities and Mathematics Proficiency
3.2.1 Attention
3.2.1.1 Attentional Abilities with Mathematics Learning Profiles
3.2.1.2 Mathematical Abilities Within Attentional Profiles
3.2.2 Executive Functioning
3.2.3 Working Memory
3.2.4 Processing Speed
3.2.5 Cognitive Load Theory
3.3 Intervention for Mathematics Remediation
3.3.1 Task-Specific Intervention and Remediation
3.3.2 Function-Specific Intervention and Remediation
3.3.2.1 Indirect Intervention of Attention-Related Skills
3.3.2.2 Direct Intervention on Attention
3.3.2.3 Direct Intervention on Working Memory
3.4 Discussion
3.4.1 Future Research
3.4.2 Implications
3.5 Conclusion
References
Chapter 4: Instructional Support for Fact Fluency Among Students with Mathematics Difficulties
4.1 Background
4.2 Typical Developmental Trajectories
4.3 Development of Fact Fluency Among Children with Mathematical Difficulties
4.4 Importance of Developing Fact Fluency
4.5 Evidence-Based Instructional Intervention Strategies for Fluency Building
4.6 Incremental Rehearsal Strategy
4.6.1 Research Supporting the Incremental Rehearsal Strategy
4.6.2 Steps for Implementing Incremental Rehearsal
4.6.3 Implications for Classroom Practice
4.7 Summary
References
Chapter 5: The Development of Arithmetic Strategy Use in the Brain
5.1 The Development of Arithmetic Strategies
5.2 Arithmetic Strategies in the Developing Brain
5.2.1 Brain Regions that Are Activated During Arithmetic Procedures
5.2.2 Brain Regions that Are Activated During Arithmetic Fact Retrieval
5.3 The Use of Educational Interventions and Experimental Paradigms to Study the Development of Arithmetic Strategies in the B...
5.3.1 Arithmetic Educational Interventions
5.3.2 Experimental Manipulations: Arithmetic Drill Studies
5.4 Discussion
References
Chapter 6: The Role of Neuropsychological Processes in Mathematics: Implications for Assessment and Teaching
6.1 The PASS Theory of Intelligence
6.2 Operationalization of PASS Processes
6.3 The Relation of PASS Processes with Mathematics Performance
6.4 Clinical Use of CAS
6.4.1 The Cognitive Profile of Children with Mathematics Giftedness
6.4.2 The Cognitive Profile of Children with Mathematics Disabilities
6.5 Interventions Based on PASS Theory
6.6 Conclusion
References
Chapter 7: The Interplay Between Motivation and Cognition in Elementary and Middle School Mathematics
7.1 Situated Expectancy-Value Theory
7.1.1 Domain-Specific Cognitive Factors and Situated Expectancy-Value Theory
7.1.2 Domain-General Cognitive Factors and Situated Expectancy-Value Theory
7.1.3 Summary of Situated Expectancy-Value Findings
7.2 Self-Determination Theory
7.2.1 Domain-General Cognitive Factors and Self-Determination Theory
7.2.2 Summary of Self-Determination Theory Findings
7.3 Achievement Goal Theory
7.3.1 Domain-General Cognitive Factors and Achievement Goal Theory
7.3.2 Cognitive and Meta-cognitive Strategy Use and Achievement Goal Theory
7.3.3 Summary of Achievement Goal Theory Findings
7.4 Combining Cognition and Motivation: An Example Using Situated Expectancy-Value Theory
7.5 Open Questions and Recommendations for Instructional Practice and Future Research
Appendix A: Review Methodology
References
Chapter 8: Design Principles for Digital Mathematical Games that Promote Positive Achievement Emotions and Achievement
8.1 Introduction
8.2 Why Are Digital Mathematical Games Effective?
8.3 Emotions
8.3.1 Achievement Emotions
8.3.2 Importance of Achievement Emotions
8.4 Emotional Foundations of Digital Game Design
8.4.1 Visual Aesthetic Design
8.4.2 Musical Score
8.4.3 Game Mechanics
8.4.4 Narrative
8.4.5 Incentive Systems
8.5 Do Emotional Design Principles Promote Positive Achievement Emotions and Learning Outcomes?
8.5.1 Review Process
8.5.2 Which Emotional Design Principles Are Used in Mathematical Game Research?
8.5.3 How Effective Are Emotional Design Principles at Improving Achievement Emotions and Learning Outcomes?
8.5.3.1 Visual Aesthetic Design
8.5.3.2 Musical Score
8.5.3.3 Game Mechanics
8.5.3.4 Narrative
8.5.3.5 Incentive System
8.6 Summary of Design Principles of Digital Mathematical Games´ Impact on Achievement Emotions and Achievement
References
Part II: Mathematical Understanding
Chapter 9: The Number Line in the Elementary Classroom as a Vehicle for Mathematical Thinking
9.1 Introduction
9.2 The Number Line
9.3 Methodological Considerations
9.4 The Instructional Sequence
9.5 Results and Discussion
9.6 Conclusion
References
Chapter 10: Longitudinal Approaches to Investigating Arithmetic Concepts Across the Elementary and Middle School Years
10.1 Introduction
10.2 The Importance of Conceptual Knowledge of Arithmetic
10.3 How to Measure Conceptual Knowledge of Arithmetic
10.4 The Development of Conceptual Knowledge of Arithmetic: Part I
10.5 Study Designs for Assessing Conceptual Knowledge of Arithmetic
10.5.1 The Cross-Sectional Design
10.5.2 The Longitudinal Design
10.6 Development of Conceptual Knowledge: Part II
10.7 A Longitudinal Study of Additive and Multiplicative Inversion, Associativity, and Equivalence
10.8 Future Directions and Practical Implications
References
Chapter 11: Obstacles in the Development of the Understanding of Fractions
11.1 Introduction
11.2 Conceptual and Procedural Knowledge of Fractions
11.3 How Natural Number Knowledge Both Facilitates and Hinders Fraction Learning
11.4 Educational Interventions and Implications for Teaching
11.4.1 The Concrete-Representational-Abstract Sequence
11.4.2 Playful and Game-Based Interventions
11.4.3 Improving Pre-service Teachers Pedagogical Content Knowledge
11.4.4 What Can Parents Do to Help Their Children Learn Fractions?
11.5 Conclusion
References
Chapter 12: The Role of Groundedness and Attribute on Students´ Partitioning of Quantity
12.1 Introduction
12.2 The Role of Problem Characteristics in Word Problem Solving
12.3 External Representations in Problem Solving
12.4 Groundedness and Equal-Sharing
12.5 Role of Object Attribute in Partitioning Strategies
12.6 An Investigation of Children´s Partitioning Strategies as a Function of Problem Features
12.7 Documenting Children´s Partitioning Strategies
12.7.1 Equal-Sharing Problems
12.7.2 Picture Perception Task
12.8 Results
12.8.1 Mental Representations
12.8.2 Partitioning Strategies
12.9 Discussion
12.9.1 The Role of Object Groundedness
12.9.2 The Role of Object Attribute
12.9.3 The Role of Unit Type
12.10 Practical Implications and Conclusion
References
Chapter 13: Designing Worked Examples to Teach Students Fractions
13.1 Introduction
13.2 Human Cognitive Architecture
13.3 The Worked Example Effect
13.4 The Worked Example Effect and Age Differences
13.5 Empirical Evidence of the Effectiveness of Worked Examples with Lower Primary School Students
13.6 Conclusion and Future Research
References
Chapter 14: Developing Fraction Sense in Students with Mathematics Learning Difficulties: From Research to Practice
14.1 Domain Specific Concepts, Procedures, and Representations
14.1.1 Why Are Fractions Hard for So Many Students?
14.1.2 Fraction Magnitude and Equivalence
14.1.3 Fraction Arithmetic
14.1.4 Common and Persistent Fraction Arithmetic Errors
14.1.5 Representations to Build Fraction Knowledge
14.2 Techniques That Support Learning Across Domains
14.2.1 Using Integrated Models
14.2.2 Connecting Concrete and Abstract Representations of Concepts
14.2.3 Using Gestures to Promote Learning
14.2.4 Distributing and Interleaving Practice
14.2.5 Providing Retrieval Practice with Corrective Feedback
14.2.6 Presenting Side by Side Comparisons to Promote Relational Thinking
14.3 Development of the FSI
14.3.1 Description of the FSI
14.3.2 Efficacy of the FSI
14.4 Concluding Remarks
References
Index
